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Let $\Gamma=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$with respect the generating set $T\not\ni0$. $\Gamma$ has orderord$(\Gamma)=|G|=n$ and degree deg$(\Gamma)=|T|=d$. Let $k(\Gamma)$ be thediameter of $\Gamma$ and denote$\kappa(d,n)=\min\{k(\Gamma):~\textrm{ord}(\Gamma)=n,\textrm{deg}(\Gamma)=d\}$.  We give a closed expression, $\ell(d,n)$, of a tight lower bound of$\kappa(d,n)$ by using the so called {\em solid density} introduced byFiduccia, Forcade and Zito.  A digraph $\Gamma$ of degree $d$ is called {\em tight} when$k(\Gamma)=\kappa(d,|\Gamma|)=\ell(d,|\Gamma|)$ holds. Recently, the {\emDilating Method} has been developed to derive a sequence of digraphs ofconstant solid density. In this work, we use this method to derive a sequenceof tight digraphs $\{\Gamma_i\}_{i=1}^{\textrm{c}(\Gamma)}$ from a given tightdigraph $\Gamma$. Moreover, we find a closed expression of the cardinalityc$(\Gamma)$ of this sequence. It is perhaps surprising that c$(\Gamma)$ dependsonly on $n$ and $d$ and not on the structure of $\Gamma$.

On solid density of Cayley digraphs on finite Abelian groups